![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > deccl | Structured version Visualization version GIF version |
Description: Closure for a numeral. (Contributed by Mario Carneiro, 17-Apr-2015.) (Revised by AV, 6-Sep-2021.) |
Ref | Expression |
---|---|
deccl.1 | ⊢ 𝐴 ∈ ℕ0 |
deccl.2 | ⊢ 𝐵 ∈ ℕ0 |
Ref | Expression |
---|---|
deccl | ⊢ ;𝐴𝐵 ∈ ℕ0 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-dec 12700 | . 2 ⊢ ;𝐴𝐵 = (((9 + 1) · 𝐴) + 𝐵) | |
2 | 9nn0 12518 | . . . 4 ⊢ 9 ∈ ℕ0 | |
3 | 1nn0 12510 | . . . 4 ⊢ 1 ∈ ℕ0 | |
4 | 2, 3 | nn0addcli 12531 | . . 3 ⊢ (9 + 1) ∈ ℕ0 |
5 | deccl.1 | . . 3 ⊢ 𝐴 ∈ ℕ0 | |
6 | deccl.2 | . . 3 ⊢ 𝐵 ∈ ℕ0 | |
7 | 4, 5, 6 | numcl 12712 | . 2 ⊢ (((9 + 1) · 𝐴) + 𝐵) ∈ ℕ0 |
8 | 1, 7 | eqeltri 2824 | 1 ⊢ ;𝐴𝐵 ∈ ℕ0 |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2099 (class class class)co 7414 1c1 11131 + caddc 11133 · cmul 11135 9c9 12296 ℕ0cn0 12494 ;cdc 12699 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-resscn 11187 ax-1cn 11188 ax-icn 11189 ax-addcl 11190 ax-addrcl 11191 ax-mulcl 11192 ax-mulrcl 11193 ax-mulcom 11194 ax-addass 11195 ax-mulass 11196 ax-distr 11197 ax-i2m1 11198 ax-1ne0 11199 ax-1rid 11200 ax-rnegex 11201 ax-rrecex 11202 ax-cnre 11203 ax-pre-lttri 11204 ax-pre-lttrn 11205 ax-pre-ltadd 11206 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-ov 7417 df-om 7865 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-er 8718 df-en 8956 df-dom 8957 df-sdom 8958 df-pnf 11272 df-mnf 11273 df-ltxr 11275 df-nn 12235 df-2 12297 df-3 12298 df-4 12299 df-5 12300 df-6 12301 df-7 12302 df-8 12303 df-9 12304 df-n0 12495 df-dec 12700 |
This theorem is referenced by: 10nn0 12717 3declth 12731 3decltc 12732 decleh 12734 decmul1 12763 bpoly4 16027 fsumcube 16028 3dvds2dec 16301 dec2dvds 17023 dec5dvds2 17025 2exp8 17049 2exp11 17050 2exp16 17051 prmlem2 17080 37prm 17081 43prm 17082 83prm 17083 139prm 17084 163prm 17085 317prm 17086 631prm 17087 1259lem1 17091 1259lem2 17092 1259lem3 17093 1259lem4 17094 1259lem5 17095 1259prm 17096 2503lem1 17097 2503lem2 17098 2503lem3 17099 2503prm 17100 4001lem1 17101 4001lem2 17102 4001lem3 17103 4001lem4 17104 4001prm 17105 slotsbhcdif 17387 slotsbhcdifOLD 17388 cnfldfunALTOLDOLD 21295 tnglemOLD 24537 quart1cl 26773 quart1lem 26774 quart1 26775 log2ublem3 26867 log2ub 26868 log2le1 26869 birthday 26873 bpos1 27203 bpos 27213 1kp2ke3k 30243 9p10ne21 30267 dp3mul10 32603 dpmul1000 32604 dpadd 32616 dpmul 32618 dpmul4 32619 hgt750lemd 34216 hgt750lem 34219 hgt750lem2 34220 hgt750leme 34226 tgoldbachgnn 34227 tgoldbachgt 34231 kur14lem9 34760 420gcd8e4 41414 12lcm5e60 41416 60lcm7e420 41418 3exp7 41461 3lexlogpow5ineq1 41462 3lexlogpow5ineq2 41463 3lexlogpow5ineq5 41468 aks4d1p1 41484 sqn5i 41781 decpmulnc 41783 decpmul 41784 sqdeccom12 41785 sq3deccom12 41786 235t711 41789 ex-decpmul 41790 sq45 42017 sum9cubes 42018 resqrtvalex 42998 imsqrtvalex 42999 inductionexd 43508 fmtno3 46814 fmtno4 46815 fmtno5lem1 46816 fmtno5lem2 46817 fmtno5lem3 46818 fmtno5lem4 46819 fmtno5 46820 257prm 46824 fmtno4prmfac 46835 fmtno4nprmfac193 46837 fmtno5faclem1 46842 fmtno5faclem2 46843 fmtno5faclem3 46844 fmtno5fac 46845 fmtno5nprm 46846 139prmALT 46859 31prm 46860 127prm 46862 m7prm 46863 m11nprm 46864 11t31e341 46995 2exp340mod341 46996 341fppr2 46997 nfermltl2rev 47006 evengpoap3 47062 bgoldbachlt 47076 tgoldbachlt 47079 ackval3012 47688 ackval41a 47690 ackval41 47691 ackval42 47692 prstcocvalOLD 48001 |
Copyright terms: Public domain | W3C validator |